Problem: Simplify and expand the following expression: $ \dfrac{5}{4p + 4}+\dfrac{-6}{2p - 5} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4p + 4)(2p - 5)$ Multiply the first term by $\dfrac{2p - 5}{2p - 5}$ $ \begin{align*} \dfrac{5}{4p + 4} \times \dfrac{2p - 5}{2p - 5} & = \dfrac{(5)(2p - 5)}{(4p + 4)(2p - 5)} \\ & = \dfrac{10p - 25}{(4p + 4)(2p - 5)}\end{align*} $ Multiply the second term by $\dfrac{4p + 4}{4p + 4}$ $ \begin{align*} \dfrac{-6}{2p - 5} \times \dfrac{4p + 4}{4p + 4} & = \dfrac{(-6)(4p + 4)}{(2p - 5)(4p + 4)} \\ & = \dfrac{-24p - 24}{(2p - 5)(4p + 4)}\end{align*} $ Now we have: $ = \dfrac{10p - 25}{(4p + 4)(2p - 5)} + \dfrac{-24p - 24}{(2p - 5)(4p + 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{10p - 25 - 24p - 24}{(4p + 4)(2p - 5)} $ $ = \dfrac{-14p - 49}{(4p + 4)(2p - 5)}$ Expand the denominator: $ = \dfrac{-14p - 49}{8p^2 - 12p - 20}$